\newproblem{lay:6_2_6}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 6.2.6}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
	Let $\mathbf{u}_1=\begin{pmatrix}5\\-4\\0\\3\end{pmatrix}$, $\mathbf{u}_2=\begin{pmatrix}-4\\1\\-3\\8\end{pmatrix}$, and $\mathbf{u}_3=\begin{pmatrix}3\\3\\5\\-1\end{pmatrix}$.
	Is the set $S=\{\mathbf{u}_1,\mathbf{u}_2,\mathbf{u}_3\}$ orthogonal?
}{
   % Solution
	To check whether $S$ is orthogonal, we calculate all possible inner products to check if they are 0 or not
	\begin{center}
	  \begin{tabular}{l}
			$\mathbf{u}_1\cdot\mathbf{u}_2=5\cdot(-4)+(-4)\cdot1+0\cdot(-3)+3\cdot8=-20-4+24=0$\\
			$\mathbf{u}_1\cdot\mathbf{u}_3=5\cdot3+(-4)\cdot3+0\cdot5+3\cdot(-1)=15-12-3=0$\\
			$\mathbf{u}_2\cdot\mathbf{u}_3=(-4)\cdot3+1\cdot3+(-3)\cdot5+8\cdot(-1)=-12+3-15-8=-32$\\
		\end{tabular}
	\end{center}
	$\mathbf{u}_2$ is not orthogonal to $\mathbf{u}_3$. Consequently, the set $S$ is not orthogonal.
}
\useproblem{lay:6_2_6}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
